2000 OPA (O!er\ed\ Pubh,hers A5wciatlonI N.V
Publ~shrdb) iiccn5e under
the Gordon and Breach Sclznce
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Printed In M a l a y w
Modelling The Effect of Cell Shedding on Avascular
Tumour Growth
J. P. WARD* and J. R. KING
Department
of'Throrerical Mechar~ics,Urliversity cf Nurtir~glzam,Nottingharn, NG7 2RD, UK
Earlier mathematical models of the authors which describe avascular tumour growth are
extended to incorporate the process of cell shedding, a feature known to affect the growth
of rnulticell spheroids. A continuum of live cells is assumed within which. depending on
the concentration of a generic nutrient, movement (described by a velocity field) occurs
due to volume changes caused by cell birth and death. The necrotic rnaterial is assumed to
contain a mixture of basic cellular material (assumed necessary for creating new cells) and
a non-utilisable material which may inhibit mitosis. The rate of cell shedding is taken to be
proportional to the mitotic rate, with constant of proportionality O. Numerical solutions of
the resulting system of partial differential equations indicate that, depending on O and the
initial conditions, the solution may either tend to the trivial state in finite time (by which we
mean complete death of the tumour), or to one of two non-trivial states, namely a steady-state
(indicating growth saturation) or a travelling wave (indicating continual linear growth). These
long time outcomes are explored by deriving the travelling wave and steady-state limits of
the model. Numerical solutions demonstrate that there are two branches of solutions, which
we have termed the 'Major' and 'Minor' branches, consisting of both travelling waves and
steady-states. The behaviour of the solutions along each branch is discussed, with those of
the Major branch expected to be stable. Beyond some critical O, where the Major and Minor
branches merge, the spheroid ultimately vanishes whatever the initial turnour size due to the
effects of cell shedding being too strong for it to survive. The regions of existence of the
two long time outcomes are investigated in parameter space, cell shedding being shown to
expand significantly the parameter ranges within which growth saturation occurs.
K e y w r d ~ Tumour
:
growth; mathematical modelling: cell shedding; numerical solution: asymptotic
analysis
1 INTRODUCTION
d e v e l o p m e n t (Folkrnan a n d Hochberg, 1973). The
Multicell spheroid cultures h a v e been u s e d exten-
penetration o f nutrients by diffusion f r o m t h e external
sively as
i r ~ vitla analogues o f t u m o u r growth,
m e d i u m into the t u m o u r spheroid, giving rise to a
mimicking the early, avascular stages o f t u m o u r
characteristic 'three-stage' pattern o f g r o w t h (namely
g r o w t h o f these cultures i s governed mainly b y the
*Corresponding Author: E-mail: John.Ward@nottingham.ac.uk
156
J. P. WARD AND J. R. KING
exponential, linear and growth saturation); see Congar
and Ziskin (1983), Inch et al. (1970) and Carlsson
(1977). A number of other factors are known to influence spheroid growth, of which the self-production
of mitotic inhibitor molecules (chalones) has perhaps been the focus of most study (see for example
Freyer et a/., 1988; Harel et al., 1984; Iversen, 1991).
However, a feature which undoubtedly has a major
effect on growth, but has been the subject of relatively little work, is the process of cell shedding. This
involves the detachment of cells from the surface of
a multicell spheroid, after which they disperse into
the growth medium; since this medium is frequently
renewed (usually about every 2 days), the result is
an irretrievable loss of the shed cells. For tumours in
vivo it has been speculated that the cell shedding process relates to cell detachment and metastasis (Weiss,
1978; Landry e f a/., 1981). However, this is uncertain, it being probable that the neighbouring tissues
will prevent the escape of many of these cells.
The shedding of cells has often been noted
in experimental reports, but there has been little
analysis. In Landry et a / . (1981), it was observed that
cells are more prone to detachment during mitosis
and it was claimed that this is due to the weakening
of the cell-cell contacts. Cell detachment during
mitosis will therefore be the focus of the modelling
in this paper. Weiss (1978) showed that exposure of
a spheroid to necrotic products from the core also
enhances cell shedding. The rate at which cells are
lost depends strongly on the cell line. Landry et al.
(1981), in their study of EMT61Ro mammary tumour
cultures, examined spheroids of diameters between
400-900 ym and estimated the cell shedding rate to
be 218 cells/mm2h, which represents 0(1%) of the
total cell content of the spheroid being lost each hour.
In Sasaki e t a / . (1984), cells were observed to be
shed from HeLa S3 rnulticell spheroids, of diameter
475 ym. at rates of about 21 cells/mm2h; this cell
shedding rate was approximately halved when these
cells were cocultured with human fibroblast cells.
Kohno ef a / . (1986) studied two cell lines and found
cell shedding rates of about 50 cells/mm2h and
5 cells/mm'h for DND 1 A melanoma and PC-I0
squamous multicell spheroids, respectively. They also
noted that the cell shedding rate was reduced during
the application of several chemotherapeutic drugs.
Such matters are not addressed here, but provide
scope for future modelling.
The process of cell shedding has been included
in few mathematical models of multicell spheroid
growth, with Landry et a/. (1 982) and Casciari rt al.
(1992) being the only known examples. In Landry
et al. (1982) the tumour was modelled as a spheroid
with an outer rim in which volume is gained through
cell birth and a core in which volume is lost by
cell death, cell shedding being modelled as a volume
sink proportional to the surface area of the spheroid.
The resulting model predicts the initial exponential
and linear phases of growth, but not the final saturation phase. Their model achieved saturation when
it was coupled with the inhibitor model of Shymko
and Glass (1976); however, the cells in the resulting saturated spheroid are fully inhibited, i.e. all cells
are quiescent, contradicting the experimental observations of Carlsson (19771, Folkman and Hochberg
(1973), Freyer and Sutherland (1986) and Haji-Karim
and Carlsson (1978), in which a surface layer of
cells continues to undergo mitosis. Casciari et al.
(1992) used a similar approach to model the cell
shedding process in their model of spheroid growth.
Their model, however, is considerably more complex
than the Landry model, the role of various molecules
involved in the cellular respiratory process being considered in order to predict the increase in pH in the
core; however, their model again failed to predict
growth saturation. The main aim of these models
was to provide good quantitative agreement for the
growth of EMT6IRo mammary multicell spheroids
using known parameter values. In the mathematical
model described below we aim to examine the role of
the cell shedding strength on both the qualitative and
quantitative behaviour, to gain a deeper understanding
of its effects on spheroid growth.
In this paper, we build on the series of models
discussed in Ward and King (1 997, 1999a, 1999b),
extending them to account for cell shedding. The
modelling approach is to assume a continuum of
live cells whose growth is driven by a generic nutrient. In the models of Ward and King (1999a) the
AVASCULAR TUMOUR GROWTH
inclusion of mechanisms of necrotic material loss,
namely leakage (via diffusion) and consumption by
living cells, enabled growth saturation to be predicted,
with the surface region being one of cell proliferation (as required by the experimental observations
noted above). The modelling was extended in Ward
and King ( 1 999b) to investigate the effects of mitotic
inhibitors released at cell death. Although the addition
of the inhibitors demonstrated little that was qualitatively new, and it was shown that the presence of
mitotic inhibitors cannot cause growth saturation on
its own, it was shown that they can have significant
quantitative effects even at low concentrations. A key
motivation of the current work was to investigate
whether cell shedding could provide an alternative
mechanism for growth saturation. The inclusion of
the cell shedding mechanism leads to modification of
the boundary conditions employed in the model of
Ward and King (199910) and it will be shown that
cell shedding has significant quantitative effects on
growth speed and saturation size and gives rise to the
existence of two non-trivial long time solutions for
a much wider class of parameter sets. A brief discussion of the model derivation together with details
of the cell shedding modelling is given in Section 2.
In Section 3, the resulting system of partial differential equations is solved numerically, with the effects
of changing the extent of cell shedding investigated.
As with the previous models, the long time outcomes
consist of travelling wave and steady-state solutions,
the travelling wave and steady-state limits being studied using both asymptotic and numerical methods in
Section 4. The bifurcations between travelling waves
and steady-states and between the existence and nonexistence of non-trivial solution are also discussed.
2 FORMULATION
2.1 Introduction
We will omit most of the details of the modelling,
since much of it is described in Ward and King
(1999b); the addition of the cell shedding mechanism
requires only the adjustment of the boundary conditions. In summary, the model of avascular tumour
157
growth is based on the assumption that living cells
form a continuum, dividing and dying at rates governed by the local concentration of a generic nutrient (e.g. oxygen and glucose) and by the availability
materials such as water, proteins and DNA required
for the construction of new cells. At cell death a
cell is assumed to dissociate into fixed quantities
of basic cellular material and non-utilisable material which may inhibit mitosis. For convenience, the
non-utilisable material will be termed an inhibitor for
the remainder of the paper. An important modelling
assumption is that the living cells, cellular material
and the inhibitive material occupy all available space
within the tumour, leading to the no void condition
where 12, y and 11 are the concentrations and VL, V,,
and Vh are the volumes of a living cell, a molecule
of cellular material and an inhibitor molecule, respectively. Volume changes due to cell birth and death and
due to the diffusion of cellular and inhibitive material
creates movement within the tunlour described by a
velocity field. The resulting sy5tem of equations is
given in non-dimensional form in Appendix A, and
in the next section the modelling of the cell shedding
process is described.
2.2 Modelling Cell Shedding
The detachment of cells from the surface, and the
consequent volume loss, implies that. in contrast to
our earlier models, the growth rate dS l d t , where S ( t )
is the coordinate of the tumour surface, is not equal
to the surface velocity. The volume loss rate due to
cell shedding is given by ~ T S ~ ( O (t)S , dS (t)/dt),
where z, is the internal spheroid velocity. The volume
lost is assumed to consist of living cells and cellular
and inhibitive material at quantities proportional to
their surface densities; defining N(t) to be the number
of cells lost through shedding, the rate at which the
surface cells are shed is thus given by
- = 457s'
'nt
"
(
"
v(S , t ) - - ( t )
nt
n(S. t ) .
(2)
J. P. WARD AND J. R. KING
158
The rate of cell shedding is assumed to be
proportional to the number of surface cells, by which
we mean the cells that occupy the space between S
and S - 2ro. where ro is the radius of cell; the number
of cells susceptible to shedding is therefore given by
n ( S , t ) 4 n ( s 3 - ( S - 2ro)"/3. We can thus write the
cell shedding rate as
dN
-= 8 . i ~ r o s ' ~ ( ~ / r on
) ~, y( ,ch, ) n ( ~t ), ,
dt
(3)
where c is the nutrient concentration,
and K ( c ,11, p , 11) is some function describing the
effects of the microenvironment on the cell shedding
rate. Eliminating d N / d t between Equations ( 2 ) and
( 3 ) results in a differential equation for the moving
boundary S ( t ) , namely
-
We note the model of Ward and King (1999b) can
0. reducing ( 5 )
be derived simply by setting K
to dS l d t = 7)(S,t ). Our continuum modelling is only
really valid for S >> vo, so it might seem reasonable
to set F ( S ) = 1; however, while this condition is typically satisfied for almost all of the growth, we shall
start from an initial state with S ( 0 ) = ro, and the full
form ( 4 ) is then needed if the cell shedding rate is
not always to be too strong for smaller spheroids;
the form ( 4 ) is thus adopted as the most appropriate
way to describe all stages of growth. Nevertheless,
as before (Ward and King, 1997) results of the continuum model on very early timescales, for which
S / r o = 0(1), should be treated with caution, as the
continuum assumption on living cells only holds if
there are a large number of cells.
In view of the observations of Landry et nl. (1981)
that cells are more prone to detach during mitosis, we
assume K ( c . n ,y , h ) to be proportional to the mitotic
rate, k,,, at the spheroid surface, so that
K ( c , n . p . h ) = O k , , ( c ( S . t ) , p ( S . t ) ,h ( S , t ) ) .
Here, the dimensionless constant O is the
proportion of the surface layer cells that are
undergoing mitosis which detach. As the spheroid
develops, so that k,(c(S t ) ,p(S , t ) , h ( S , t ) ) becomes
approximately constant, this choice for function K
approaches the cell shedding term used by Landry
et al. (1982) of cell shedding rate being proportional
to the surface area. We note also that defining
K ( c , n ,p , h ) in this way means that the model for cell
shedding contains only one more parameter than the
model of Ward and King (1999b), namely O, values
for which can be obtained from the literature.
Experimental data is usually expressed in terms of
the rate of cell shedding per unit area ( R c s ) , which
is simply ( d /dt)/4.irs2,
~
so from ( 3 ) we have
is the mitotic rate at the spheroid surface.
where if,,
Equation ( 6 ) can be used to determine suitable values
for O from existing data. For example, using the
cell shedding rate of 218 cells/mm2h given in Landry
et al. (1981) and the data listed in the Appendix of
Ward and King (1997), whereby n ( S . t ) = 1/VL =
1 o9 cm-" ro = 6 x 10p\m, if,,
=A = IO-~S-' (A
being the maximum mitotic rate) and F ( S / r o ) = 1
(which holds for S >> ro) we have O = 0.5. 8 will
be varied in this paper to obtain an understanding of
its effects on growth.
The difference between growth rate and surface
velocity in ( 5 ) also requires a modification of the
Robin boundary conditions used in Ward and King
(1999b) to describe the cellular and inhibitive material flux at the spheroid surface. The cell shedding
modification gives at r = S ( t )
where Q , and Qh are the mass transfer coefficients
and j)g and ho are the external concentrations of the
cellular and inhibitive material, respectively.
The full system of dimensionless equations
implementing the cell shedding process is given in
Appendix A. Before discussing numerical solutions
of the model we first describe special cases relating
the current model to previous works and then
AVASCULAR TUMOUR GROWTH
introduce classifications of the long time solutions
which are of relevance to the later sections.
It was found necessary in some of the simulations
described later to allow the spheroid to grow without
cell detachment before switching on the cell shedding
mechanism, to enable investigation of non-trivial
large time outcomes by preventing extinction of the
spheroid when it is small in size; as we shall see, an
important effect of incorporating cell shedding is to
cause spheroids of less than some critical size to cease
to be viable. We note that it is possible that cell lines
with inherently high cell shedding rates, corresponding to very low cell-cell adhesion, may be unable to
form spheroids.
2.3 Special Cases
The cell shedding model is a generalisation of the
models described in Ward and King (1997, 19993.
and 1999b). The following are also special cases and
will be discussed later in this paper. We note that
all quantities discussed here are dimensionless and
described in more detail in Appendix A.
159
the cellular material) and u = 1 (preventing volume
loss through inhibitor breakdown). It will be shown
in Section 4.2 that, except in the case of 6 = 0
where no inhibitor is produced, only travelling wave
solutions or tumour extinction can then result in
large time.
No leakage model
This is derived by setting Q,, = Q,, = 0. It will be
shown that the tumour dies out for all 0 0 if
b - ( 1 - 11)p > A.
>
2.4 Classification of Long Time Solutions
For the case O = 0, it was shown in Ward and King
(1999a) that there may be zero, one or two branches
of travelling wave and steady-state (i.e. non-trivial)
solutions, depending on the parameter values. It will
be convenient below to use the terms Regime I and
Regime 11, introduced in Ward and King (1999b), and
defined as follows.
Regime I
Cell shedding-no inhibitor model
For this model we set the parameters so that there
is no production or external source of the mitotic
inhibitor, i.e. we take / I = 0 and ho = 0.
Parameter sets (with 0 = 0) for which only one
branch of non-trivial long time solutions exists. These
occur when a ( I , yo, ho) > 0.
Regime I1
Cell shedding-leakage only model
Here there is no cellular material consumption or
inhibitor production, so that leakage is the only mechanism for necrotic volume loss. This is achieved by
setting y, = 0 and X = 0, preventing utilisation of the
cellular material during mitosis. and p = 0 and ho = 0,
so that there is no source of inhibitor. As with the
Leakage only model of Ward and King (1999a) these
conditions may lead to either growth saturation or a
travelling wave.
Cell shedding-inhibition model
This is obtained by setting Q,, = Qh = 0 (preventing leakage). X = y, = 0 (preventing consumption of
Parameter sets (with O = 0) for which either two or
zero branches of non-trivial long time solutions exist.
These occur when a ( l , p o .ho) < 0.
These regimes can be identified by, for example,
studying the travelling wave and steady-state limits of
Equations (20)-(22) in the limit of X + x;see Ward
and King (199%. 1999b). Defining 110 by a(1. 1 no - ho, ho) = 0, it can be shown for 0 = 0 that if
170 +yo + 11,~
I then the long time solutions lie in
Regime I, otherwise the solutions lie in Regime 11;
the inequalities above follow because a ( r , y > / z )is
monotonic increasing in p . Regime I is relevant for
all A, but non-trivial Regime I1 solutions arc restricted
to a finite range of A; which case applies depends on
the values of the other parameters.
>
160
J. P. WARD AND J. R.KING
For 0 > 0 the situation is rather different. As
will be apparent from the studies of the long time
behaviour in Sections 3 and 4 below, the addition of
the cell shedding term leads to the existence of two
non-trivial branches of travelling wave and steadystate solutions which, like Regime I1 solutions, can
be characterised by the ultimate size of the spheroid.
A typical example is given in Figure I , where the
curves are generated from numerical solutions to
the systems derived in Section 4. The two branches
of long time solutions are illustrated by the solid
and dashed curves in the figure; we expect that the
dashed solutions are unstable to radially symmetric
perturbations, and this is discussed in the Appendix B
for the limit 0 4 0 with S = 0(0'!'). To the left of
the dotted line (which is the travelling wavelsteadystate bifurcation on the upper branch), both travelling
wave (not shown) and steady-state solutions exist,
whilst to the right, both long time solutions are
steady-states. To distinguish between these branches.
it is convenient to introduce the terms 'Major' and
'Minor' branch. as follows.
Major branch solution
As 8
+ 0, these solutions tend smoothly to the
non-trivial 0 = 0 solution (under Regime I) or the
upper branch O = 0 solution (under Regime 11). The
long-time growth velocity and saturation size are
monotonic decreasing in 0 : at the turning point the
Major and Minor branches meet. These solutions are
expected to be stable.
Minor branch solution
As O i 0, these solutions tend smoothly to the trivial solution (under Regime I) or the lower branch
0 = 0 solution (under Regime 11);an example of this
behaviour will be given in Section 4. It appears that
the long time solution on this branch is always a
steady-state. with the saturation size n~ostlyincreasing in O (until the turning point is reached). There is
the possibility of there being additional folds along
this branch depending on the form of function F ( S )
in Equation (4); an example of this behaviour is discussed in Section 4.3. These solutions are expected
to be unstable.
These branches meet at some critical value of O =
O,, with no non-trivial long time solutions existing
for O > 0,. due to the cell shedding rate being too
strong, so that the spheroid eventually vanishes.
We note that the curve of saturation size against
O illustrated in Figure 1 is qualitatively similar to
that in Figure 3 of Byrne and Chaplain (1996), where
saturation size is plotted against a term proportional to
the surface tension per unit surface area, 7 . However.
an increasing O can be thought of as corresponding
to reduction in cell-cell adhesion forces, the physical
FIGURE 1 Figures showing the effects of the cell shedding factor O on the saturation (steady-state) spheroid radius. The solid and
dashed curves respectively illustrate the Major and Minor branch long time solutions, and the dotted curve indicates the travelling
wavelsteady-state bifurcation.
AV.4SCULAR TUMOUR GROWTH
mechanisms responsible for the fold are therefore
quite different.
3 NUMERICAL RESULTS
The numerical procedure used to approximate the
system (20)-(23) subject to (26) is essentially the
same as that described in Ward and King (1999a). The
system is rescaled to fix the moving boundary to unity
using I. = S ( t ) p and is solved using finite differences
in a predictor-corrector type fashion, NAG routines
being used on Equations (20). (21) and (22). Full
details are given in Ward (1997).
Greater insight into the effects of cell shedding is
gained from the study of the long time system, so
only a few examples of the transient behaviour are
given here. The basic set of parameters used are given
by (1 9) and (20) in Ward and King (1999b), namely
Q,, = 10.
161
h,=0.1,
ho=O.
(7)
with 0 being varied. As described in Ward and King
(1999b), these parameters are based on a combination of experimental data and suitable estimates. In
particular, there is assumed to be no volume loss at
cell death (6 = 1) and the necrotic material consists of
90% cellular and 10% inhibitive materials (implied by
p = 0.1). The diffusive and leakage properties of the
cellular and inhibitive material are taken to be equal.
The external medium contains some cellular material (po= 0.1) but no inhibitor (ha = 0). The inhibitive
material can at most reduce the mitotic rate by 3 0 7 ~
( P = 0.3) and can be completely converted by the
living cells to cellular material (u = 1).
Figure 2 show the effects of 0 on spheroid growth
and on cell population loss rates, using the data given
by (7); the curve for O = 0 in Figure 2 is the same as
that shown in Figure 1 in Ward and King (19991-3). In
these simulations the cell shedding term was switched
on at t = 25. so that the spheroid had reached a sufficient size for growth to continue, where possible,
to a non-trivial long time solution. The switching on
of the cell shedding term in this manner is probably
not unreasonable, as the population reached by t = 25
reflects the initial population of experimentally grown
FIGURE 2 Tumour radius against time for various values of the cell shedding parameter 8 . The curves coinicide until t = 25, when
cell shedding is turned on.
J . P. WARD AND J. R. KING
162
multicell-spheroids which is typically about lo4 cells.
The figure demonstrates that the saturation size is
strongly dependent on O and, as expected, increasing O reduces the spheroid saturation size. Beyond a
bifurcation value of O,
1.46 (see next section) the
cell shedding rate becomes so strong that the spheroid
eventually vanishes (an example of this being given
for O = 1.5). Calculating the rates of cell population
loss per unit area (Rcs) using Equation (6), reverting to dimensional quantities (using A = 1 0 - ~ s - ' ,
ro = 6.2 x lop4 cm; see the Appendix in Ward and
King, 1997), comparison can be made between these
results and experimental results from the literature. In
this simulation we find at growth saturation that Rcs
is about 1.0. 1.9 and 3.0 cells/cm2s for O = 0.25,0.5
and 1, respectively, these being in line with the experimental results of Landry et al. (1981, 6 cells/cm2s).
Sasaki et al. (1984, 0.6 cells/cm2s) and Kohno ef al.
(1986, 1.4 and 0.13 cells/cm2s).
In Figure 3 the spheroid radius is plotted against
time for the Cell shedding-leakage only case, using
the parameter set (7) but with D, = 800 and Q, =
100, so that the curve for O = 0 is the same as the
curve for D,, = 800 in Figure 11 of Ward and King
( 1 999a). For O > 0, cell shedding was switched on at
-
f = 10, again to give the spheroid a chance to grow.
For O = 0 and 0.25 growth ultimately becomes linear,
whilst for O = 0.5,0.75 and 1 it saturates, the bifurcation occurring at O,. x 0.39. We note that growth
again tends towards solutions on the Major branch
in these examples. The decrease of both the linear
growth speed and the saturation size on increasing
(3 is to be expected, and this figure demonstrates
how the inclusion of cell shedding can induce growth
saturation when it would not otherwise occur. It can
be observed numerically that the cell loss rate per
unit area levels off in all cases, with rates Rcs of
2.5, 5.1, 7.6, 10.1 cells/cm2s for O = 0.25, 0.5, 0.75,
1. respectively. again conlparable with experimental
results.
An investigation into the stability of a particular Minor branch solution is illustrated in Figure 4,
using the smaller of the two steady-state solutions
(SG < ,s
: these denoting the saturation sizes on
Minor and Major branches, respectively) of the Cell
shedding-leakage only model (found by solving the
appropriate system from the next section), with parameter values given by the first three rows of (7)
and with X = 0.8, Dp = Qp = 300 and O = 0.8. The
steady-state was used as the initial condition, except
FIGURE 3 Turnour radius against time for various values of the cell shedding parameter O, without inhibition or mitotic contraction.
The dashed lines indicate the saturation radii. the saturation size for O = 0.5 being S
, z 1339. Growth does not saturate for O = 0 or
Q = 0.25.
AVASCULAR TUMOUR GROWTH
time (1)
FIGURE 4 Spheroid radius against time for various initial sizes: S ( 0 ) x 1.35 (curve A), S ( 0 )
(curve C).
that the initial size was varied and the dependence on
r scaled accordingly. The curve labelled B starts with
the actual steady-state solution (with S$ = S(0) E
2.35), and there is negligible deviation from this initial state. However, perturbing the initial size above
or below, as shown by curves A and C, leads to
growth diverging away from the curve B solution,
the smaller spheroid vanishing in finite time and the
larger growing towards a Major branch steady-state
solution (SZ E 128.6). We note that this extinction
in finite time is in contrast to the decay in infinite
time of solutions under a Regime I1 parameter set
without cell shedding (cf. Figure 10 of Ward and
King, 1999a). This finite time extinction behaviour is
predicted by the analysis in the Appendix B, which
demonstrates that Minor branch solutions are unstable
in the limit O + 0; it is expected that solutions on
this branch are always unstable.
4 LONG TIME BEHAVIOUR
-- 2.35
(curve B ) and S ( 0 )
-
3.35
U ) and steady-state (of a priori unknown saturation
size S,). The form of the nonlinear diffusion term in
(20) and (22) leads to the possibility of steady-state
solutions with fully developed (n E 0 for r < R, <
S,, for some R,), as well as a partially developed
(n > 0 for all r), necrotic cores. The long time
systems of equations for each case are listed in Ward
and King (1999b) and are omitted for brevity.
4.2 Existence of Non-trivial Solutions
In this section we examine the existence of non-trivial
solutions for the Cell shedding-inhibitor model and
the No leakage model presented in Section 2.3. For
the Cell shedding-inhibitor model, it will be demonstrated that, except in a special case, steady-state
solutions do not occur. However, unlike our earlier
models, steady-state solutions of the No leakage
model exist over a range of parameters rather than
only along a three-dimensional 'surface' in (6,A, p, u)
parameter space; non-trivial solutions exist only in a
bounded region in parameter space, defined below.
4.1 Long Time Equations
As with the model of Ward and King (1999b), the
possible long time outcomes of Equations (20)-(22)
are travelling wave (with a priori unknown speed
4.2.1 Cell shedding-inhibitor model
Recalling that for the Cell shedding-inhibitor model
we have Q, = Q h = A = p , = O and u = 1, the
J . P. WARD AIVD J. R. KING
164
travelling wave equations for n and v can be
written as
[(u - U)nI1= (k,,, - k(1)rz
U'
= (k,,, - ( I - h)k,i)rz - D, (n"
+ htl)
+ Dl,h ",
(14)
from which we can deduce
subject to D,(rz1(O) + hl(0)) = (u(0) - U ) ( l - n(0) h(0)) and Dl, = v(O)h(O). On integration this yields
0
U = 6J-x k,rnd,-. Positivity of the wave speed
implies that travelling wave solutions exist only if
6,the ratio of dead to live cell volumes, is greater
than zero. Repeating the analysis on the steady-state
equations results in h
r'k,indr = 0, implying that
non-trivial solutions exist only if S = 0. In summary
we have
6 > 0 : Only travelling wave solutions exist.
6 = 0 : Only steady-state solutions exist; these are
non-unique, being dependent on the initial
conditions (see Section 5.1 of Ward and
King, 1999b, for full details).
using (8)-(10) with Q, = Q,, = 0. In the case of
= 0, that is the inhibitor is not broken down,
the results to follow are the same as for u = 1.
With v(0) > U and (I - v ) h ( - w ) 5 1 the left-hand
side is positive, so positivity of the right-hand side
implies that
6 - (1 - u ) p > X
(15)
( 1
is a necessary condition for the existence of travelling
wave solutions in this no leakage case. Equation ( 1 5)
states that the total amount of cellular material that
can be produced through cell death (6 - (1 u)p)
needs to exceed the amount consumed at cell birth
(A). A condition for a steady-state solution can
be derived in a similar way, resulting in expression
-
soS'
Thus the material leakage and/or consumption
mechanisms are essential for growth saturation to be
predicted across a general parameter range.
4.2.2 No leakage model
Recalling that for this limit of the model we have
Q, = Qh = 0, SO using (14) together with
we may deduce for the case
1,> 0
where the inclusion of cell shedding in the model
implies that u(0) > 0 and A > 0, so that the left-hand
side is positive, implying that (15) must again hold for
a non-trivial steady-state to exist. In the case of A = 0,
i.e. zero cellular material volume, the left hand side of
(16) is zero, implying that for steady-state solutions
we must have 6 = (1 - u)p; since S p (1 - u)y,
this can hold only for the very special case 6 = /L and
u = 0 (that is, if all necrotic material is inhibitive and
is completely broken down). Numerical investigations
indicate that there are two branches of solution, with
the Major branch consisting of travelling wave and
steady-state solutions and the Minor branch of steadystate solutions only. In the earlier models, steady-state
solutions were restricted to the line S - (I - u)p = A;
now, with the addition of the cell shedding term, they
exist over a range of parameter values. In summary
we have
X > 0 : Travelling waves and steady-states solutions
exist over a range of parameters provided
that S - (1 - u j p > A.
X = 0 : Travelling wave solutions exist provided
that 6 - (I - u)p > 0. Steady-state solutions can only exist if S = p and u = 0,
in which case they are non-unique, being
dependent on the initial conditions.
> >
165
AVASCULAR TUMOUR GROWTH
From (15), we note that without the presence of
inhibitor ( p = O), steady-state solutions exist in the
range 6 > A. Moreover, unlike the model of Ward and
King (1999b). (15) is not a strict bound, the numerics
indicating that the range in which only trivial long
time solutions exist is larger than 5 - (1 - u ) p 5 A.
where = diz / d y and Qi = dh /dy , and for each of
the steady-state and bifurcation cases the modified
boundary conditions are
~ ( 1 )= 2 0 k , , ( 1 , 1 - NI
$1)
= [Q,,(l -pi)
-
4.3 Numerical Results
The derivation of the long time equations and boundary conditions and the numerical procedures are
described in Ward and King (1999b). In each case
the problem is formulated as a two-point boundary
value problem and rescaled to a coordinate y defined
on the unit domain. The only alterations to the travelling wave system given in Ward and King (1999b)
concern the boundary conditions
-
Nl
-
H I .H1)NIF(S,);
-
H I ) - 71(l)(l - N I
~ l ) l / D [ l- @ ( I ) ,
@(I) = [ Q I , ( ~-o HI)+ L'( 1)HII/DI,;
where the constants N1 = rz(l), H I = h(1) and U need
to be determined as part of the solution. The boundary
value problems are solved using the shooting and
matching technique incorporated in the NAG routine
D02AGF. The continuation procedure described in
Ward and King (1999a) is used for the investigations
in parameter space. Apart from the set of parameters
used for Figure 11, all of the results to follow use
parameter sets that lead to Regime 1 solutions when
0 = 0.
Figures 5-8 study the effects of the parameter O
on various aspects of the long time growth behaviour.
using the same data used for Figure 2. The existence
of two possible long time outcomes is demonstrated
in Figure 5, where the solid and dashed curves are
the Major and Minor branches, respectively. In this
FIGURE 5 Effects of the cell shedding factor O on spheroid saturation sire (solid SZ and dashed Sg) and nccrotic core radius
(dots), defined as the position of the necrotic interface (R,). The position of the fully/partially necrotic core bifurcation is indicated
by the o. The solid and dashed curves are the Major and Minor branches, respectively.
J. P. WARD AND J. R. KING
10
8
2
=
-
6 -
2
-e
.-6
4
-
2
-
LO
0
.'
0
0.2
0.4
1
0.8
0.6
cell shedding factor (Q
1.2
I .4
FIGURE 6 Plots of the Minor branch saturation size against Q, where the solid and dotted curve are asymptotic and numerical
solutions, respectively, using F ( S ) = 1 - 2/S + 4/3.Y2. The dashed curve is the asymptotic solution using F ( S ) = 1. The asymptotic
0 and D,, D,, + oc and the numerical (dotted) curve is a blow-up of part of Figure 5.
limit is 01, 1%
-
0.035
003
-
0.005 -
,
0
0
0.2
0.4
0.6
0.8
-____----
___-.-.--- - 7 -.----
cell sheddmg factor (Q
1
1.2
1.4
1.6
FIGURE 7 Cell population loss rate (cellds) as a function of the cell shedding factor O. The o denotes the fully/partially necrotic
core bifurcation. The solid and dashed curves are the Major and Minor branches, respectively.
case, the MajorIMinor branch bifurcation occurs at
0 z 1.46; for values of 0 beyond this the cell shedding term is too strong for there to be any non-trivial
solutions. The solid curve illustrates the (expected)
effect that the saturation size decreases with increasing 0, the fully necrotic core solutions to the left
of the '0' being larger than the partially necrotic
solutions. In this case, the Minor branch consists only
of partially necrotic solutions and shows the opposite
behaviour of saturation size increasing with 0. before
reaching the MajorIMinor branch bifurcation. Since
the Minor branch is unstable this behaviour suggests
that as O increases there is an increase in the initial
'nucleation' size a spheroid must be in order for it to
AVASCULAR TUMOUR GROWTH
0
0.4
0.2
0.6
0.8
1.2
1
cell sheddtng factor (0
1.4
1.6
FIGURE 8 The effects of the cell shedding factor 8 on the cell loss rate per unit area (in cells/cm2s). The solid and dashed curves
are the Major and Minor branches, respectively.
attain a non-trivial long time solution. This is again
to be expected, as is the existence of such a critical spherical size: the cell shedding rate depends on
the surface area (and hence S2) whereas in the early
stages of growth the total mitotic rate depends on the
volume (and hence s 3 ) ; the former dominates if S
is sufficiently small and the latter if it is sufficiently
large. We note in Figure 5 that up to about 8 = 0.75,
at least, the lower branch lies in the range of tumour
radius for which a continuum model is not valid; in
this regime the predicted 'nucleation' size will be
only a few cells (at most) so in practice spheroids
of any initial size would be expected to reach the
Major branch radius S
: in this regime.
Seeking solutions along the (non-physical) Minor
branch as O + 0, S g -+ 0 (i.e. in Regime I), we find
that c
1, h -- ho, n -- 1 -PO - ho and
-
_
.,
s,
',
I
@p ( g k d , PO.ho))
a ( l , p o , ho)
(I7)
using F(S) -- 4 / 3 s 2 as S -+ 0. We note that positivity of O implies that we need a ( l , y o ,ho) >
0, i.e. the birth rate must exceed the death
rate. For O = 0, a ( l , p o ,ho) = 0 marks the bifurcation between Regime I ( a ( l , y o ,ho) > 0) and
Regime I1 ( ~ ( 1 . ~ho)
0 , < 0) solutions, as discussed
in Section 2.4. An example of the behaviour using
a parameter set giving Regime I1 solutions for O = 0
is shown later. Greater insight into the nature of the
Minor branch can be gleaned by examining the case
of small consumption rate (PI,p2 i 0) and large diffusivities (Dp,Dh + m) for S: = O(1); this limit is
relevant to Figure 5 in which PI = 0.01, P2 = 0 and
D,, Dh = 300. Omitting the details of the analysis,
it can be shown that for S: = O(1) we have c
1,
n ii(S$), h
~ ( s z ) , u ii(S:)r/3
and
N
-
-
-
where ii(S2) = ~ ( 1 . 1 ii(Sm)- h ( s 2 ) . ~ ( s c ) ) and
?
, ( ~ ) = k ( , l - ( s - h ( ~ ) , ( ) ) . The
asymptotic solutions for this limit are compared with
the numerical solution in Figure 6, demonstrating that
the analysis predicts the slight fold seen in the numerical curve. Also shown in Figure 6 is the asymptotic
solution for F ( S ) = 1, showing that similar qualitative results occur for larger S;; the folds in the
other curves result from the behaviour of F ( S ) =
1 - 2/S + 4 / 3 s 2 for S = O(1) and are not of physical
significance.
Using Equation (4) and converting to dimensional
quantities, as before, the cell population loss rate (in
cellts) as a function of O is shown in Figure 7, the
J. P. WARD AND J. R. KING
168
Major and Minor branches forming a closed loop. TO
the left o f the rnaximurn on the Major branch, the
increasing proportion o f detached cells results in the
increase in the shedding rate. However, as O becomes
larger, causing the reduction in size o f the spheroid
and consequently o f the number o f cells at the surface, there is a decrease in the cell loss rate towards
the MajorIMinor branch bifurcation. In Figure 8 the
cell loss rate per unit area is monotonically increasing in O for both Major and Minor solutions, as is
to be expected. Again, we have converted to dimensional quantities, so that a comparison can be made
with experimentally observed values, ranging from
0.1-6 cells/cm2s (Sasaki et al., 1984; Kohno et al.,
1986; Landry et al., 1981). In the limit o f O + 0
with S: + 0 (see (17)), the cell flux at the surface
,
the rapid inican be shown to be O ( @ ' 1 3 )explaining
tial increase on the Minor branch. The abrupt increase
further along this curve corresponds to the fold in
Figure 6.
The paths o f the non-trivialltrivial (solid curve),
travelling wave/steady-state (dashed curve) and
fully/partially necrotic core (smaller-dashed curve)
bifurcations in O and D , = Q , parameter space are
shown in Figure 9 for the Cell shedding-no inhibitor
model. Here, the parameters are given by (7), except
that p = 0 so that there is no inhibitor present, and we
set A, = 0.8 (rather than X = 1 ) so that the cross-over
o f the bifurcation curves is emphasised. The solid
curve was generated by tracking along d O / d S , = 0,
which is where the Major and Minor branches meet,
as D, = Q , is varied. The solid curve is started at
about D,, = 70, due to the failure o f the numerical
procedure for smaller diffusivities (the evaluation
o f the curve dQ,/dS, = 0, i f it exists, being very
sensitive to the initial guess there). However, it is
believed that for smaller diffusivities the solid curve
tracks along the dashed curve. Below the dashed
curve, the Major branch consists o f travelling wave
solutions. To the right o f the cross-over point o f the
dashed and smaller-dashed bifurcation curves, fully
necrotic core solutions exist between those curves
and are Major branch solutions. The fully necrotic
core solutions to the left o f the cross-over point lie
above the smaller-dashed curve and are Minor branch
solutions, except for those that exist above the dashed
curve in the vicinity o f the cross-over point.
Figure 10 shows the effect o f O on the saturation
size along the dotted paths labelled A and B
in Figure 9 . The Major branch along path B
descends rapidly from the travelling wavelsteadystate bifurcation on increasing O, until it reaches the
3
25
-
2
-
15
-
Two Non-Trivial Solutions
1
-
05
-
0
1
0
1
50
100
150
200
250
300
350
400
dlffuston and mass transfer coefficient of cellular materlal (D;,= Q,J
450
500
FIGURE 9 The paths of the travelling wavelsteady-state (dashed) and fullylpartially necrotic core (shorter-dashed) bifurcations in
Dp = Qp and O parameter space. The solid curve tracks the path of dO/dS, = 0, i.e. the bifurcation between the Major and Minor
branches, above which only trivial solutions exist. The paths for Figure 10 are indicated by the dotted lines labelled A and B.
AVASCULAR TUMOUR GROWTH
FIGURE 10 Saturation size (qolid and dashed curve) against O along paths A and B of Figure 9. Major branch solutions are depicted
by the solid curve and Minor branch solutions by the dashed curves. Travelling wave solutions (not shown) exist to the left of dashed
lines at the top of the figure, marking the location of the travelling wavelsteady-state bifurcation. The fully necrotic core solutions
occuring for O between the dotted line and the o marking the fullyipartially necrotic core bifurcation.
MajorMinor branch bifurcation at 8, x 2.53. As in
Figure 5, the Minor branch consists only of partially
necrotic steady-state solutions, with saturation size
increasing in O to S z zz 33. Along path A all steadystate (including fully necrotic core solutions) exist
on the Minor branch, on which the saturation size
is monotonically increasing in 0 , blowing up as
O approaches travelling wavelsteady-state bifurcation
point. We note that the travelling wave solutions exist
to the left of travelling wavelsteady-state bifurcation
point indicated by the dashed lines in Figure 10 and
are Major branch solutions.
The saturation radius is plotted against O in
Figure 11 for a parameter set giving Regime I1
solutions if O is set to zero. The parameters are
given by (7) with X = 0.4, D,, = Q, = 250, p = 0
(SO that there is no inhibitor present) and po = 0,
implying a ( l , p o ;ho) < 0. Here, the two saturation
sizes satisfying O = 0 correspond to those occurring
for X = 0.4 on curve B in Figure 18 of Ward and
King (1990a). The figure shows that the Minor
Branch solution curve tends to the lower of the two
Regime 11 solutions in the limit O -+0 (rather than
towards the trivial solution, i.e. unlike Figures 5,
10). This is consistent with the analysis leading
to Equation (17), demonstrating the non-existence
of non-trivial solutions with SE + 0 in the limit
O + 0 for parameter sets that give rise to Regime I1
solutions when O = 0. The Minor branch solutions are
again expected to be unstable, with the Major branch
and trivial solutions being the long-time attractors.
In Figure 12 we examine the effects of O on
the travelling wavelsteady-state bifurcation curves in
D , = Q , and X parameter space. This continues a
study in Ward and King (1999b), the parameter set
being given by (7), except for D17= Q17= 400, P =
0.9 and p = 0.25. The increase in the cell shedding
flux that results from increasing O can be seen to lead
to a decrease in the number of cells in the spheroid,
thus shrinking the travelling wave regions. The nonmonotonicity of the curves is due to the shifting of
the dominant cellular material loss mechanism from
consumption to leakage on increasing D , = Q , .
5 DISCUSSION
The process of cell shedding has largely been ignored
in multicell spheroid growth models, despite the
weight of evidence that suggests it is an important factor in avascular growth, at least irz vitro
J. P. WARD AND J. R. KING
FIGURE 11 Saturation size (solid and dashed curve) and necrotic core radius R, (smaller dashed curve) against 0, for a parameter
set of Regime I1 when O = 0. The fully necrotic core solutions lie on the solid line for O between 0 and the o.
FIGURE 12 The effects of the cell shedding factor 0 on the travelling waveJsteady-state bifurcation in D,, = Qp and X space
(Landry et al., 1981; Kohno ct al., 1986; Sasaki r t al.,
1984; Weiss, 1978). In this paper the model of Ward
and King (1999b) has been adapted to include a
cell shedding term so that its effect can be investigated. Numerical results shown in Section 3 and
4 demonstrate that the cell shedding process has
significant quantitative effects, even for fairly weak
shedding rates (see, for example, Figures 2 and 5).
Furthermore, as shown in Figure 3, the qualitative
behaviour may be affected too, the addition of cell
shedding leading to growth saturation (as opposed to
continual growth) for sufficiently high cell shedding
rates (i.e. sufficiently large (3). The model continues to mimic successfully experimentally observed
growth behaviour and tumour heterogeneity when cell
shedding is included.
AVASCULAR TUMOUR GROWTH
The choice of cell shedding rate function
K ( c ,n ,y , h ) = Ok,,,( c )means that the mechanism can
be modelled with the addition of only one extra
parameter 0 : values for which can be obtained from
published experimental data (indeed, it is one of the
quantities for which good experimental information
is available, making an assessment of its effects on
the behaviour predicted by the model particularly
worthwhile). The addition of the cell shedding term
gives rise to two non-trivial branches of long time
solutions, which vary smoothly with O to form two
curves we have termed Major and Minor branches.
In our earlier models, the existence of two nontrivial long time solutions was only observed in rather
restricted parameter regimes. Beyond a critical value
O,, where the Major and Minor branches meet, there
are no non-trivial solutions due to the cell shedding
rate being too strong, forcing spheroid extinction;
cell lines with O > O, will therefore be unable to
form spheroids. The effect of strengthening the cell
shedding rate on the (stable) Major Branch solutions
is to decrease the travelling wave speed or saturation
size of the spheroid and to expand the parameter
regimes in which saturation occurs. Along the Minor
branch, consisting only of steady-states solutions, the
saturation size increases with O; these solutions are
expected to be unstable and this behaviour reflects
the need on increasing O for a larger initial spheroid
size in order for the spheroid to be viable. The
expected instability of the Minor branch solutions is
confirmed by the stability analysis in Appendix B and
is illustrated in Figure 4. We thus expect the long
time solutions on the Major branch to be the physical
ones. We note that in regions in which only trivial
long time solutions exist the cell shedding rate is
too strong for the cells to grow as spheroid cultures.
Analysis of the long time equations demonstrates
that, except in a physically implausible case of zero
dead cell volume (6= O), the process of cell shedding
and mitotic inhibition together cannot cause growth
saturation; i.e. some non-negligible level of leakage
or consutnption of cellular material is required for
saturation to be achieved. In summary, the analysis
demonstrates that necessary conditions for growth
saturation to occur are either:
171
i ) Leakage of cellular material and/or inhibitor
(i.e. Q,,, Qh > O),
or ii) Cellular material consumption and cell shedding (i.e. A, O > 0).
We note from ii) that, in contrast to our previous
models, steady-state solutions do exist over a range
of parameters in the case when the cellular material
is prevented from leaking out. We note further that,
since K(c, n , p , h ) must be positive to be biologically
appropriate, conditions i ) and ii) both hold for any
suitable choice of K(c, n , p , Iz).
As already indicated, for experiments in which
the cell shedding rate can be determined, the value
of O can readily be calculated using Equation (6),
the values of n ( , S , t ) , i,,,and ro being comparatively easy to determine experimentally. An interesting experiment would be to study spheroid growth in
a medium containing some chemical that promotes
cell shedding, so features such as growth rate and
saturation size can be determined as a function of
the cell shedding rate; an example of such a chemical is trypsin, which is often used to dissociate cells
from spheroids (for example Freyer and Sutherland,
1980). Comparison of such experiments with the
model predictions would provide a valuable (quantitative) means to assess the modelling assumptions.
All the comments above regarding stability
concern stability to radially symmetric perturbations.
There is currently (e.g. Greenspan, 1976; Chaplain
and Sleeman, 1993; Byrne and Chaplain, 1996)
widespread interest in issues of stability to non-radial
perturbations (with reference to the formation of
'hot spots' for example; see Chaplain, 1995). We
therefore also note that cell shedding is likely to be a
stabilising influence on such non-radially symmetric
perturbations, since if a part of the surface grows
more rapidly than surrounding regions the increase
in surface area there will lead to a local enhancement
of cell shedding.
We have shown that the inclusion of even fairly
low levels of cell shedding leads to quite significant
changes in the quantitative behaviour, from which
we conclude that in order to simulate the growth
of multicell spheroids accurately the effects of cell
shedding must be considered. Although it has been
172
J. P. WARD AND J. R. KING
shown that cell shedding is an important cell loss
mechanism in vivo, the situation in vivo is unclear. It
is anticipated that cells detached from a tumour body
will to a large extent be prevented from moving far by
neighbouring normal cells, and will thus remain in the
locality of the primary tumour and could conceivably
rejoin it. The number of cells that escape permanently
via the vasculature or via lymph vessels. thus being
able to form secondary growths (metastases), is likely
to be relatively low. The mathematical modelling of
the behaviour in vivo of tumour cells after they have
been shed thus presents a significant, and worthwhile,
challenge.
The changes in volume by cell birth and death and by
the diffusion of cellular and inhibitive material causes
movement within the tumour which is described by a
velocity field, leading to the convective terms in the
equations below. Using these assumptions and adopting spherical symmetry, the full system of dimensionless equations is
Appendix A: Model equations
A full description of model derivation and nondimensionalisation is given in Ward and King
(1999b); we give only a brief summary here.
The main variables in the model are live cell
density n , cellular material concentration p, inhibitor
concentration h , nutrient concentration c, velocity
1; and the moving boundary marking the tumour
boundary S , the independent variables being time t
and the radial coordinate r . The model is presented
below in non-dimensional form; it is worth noting
that time is scaled with the maximum rate of
cell birth and space is scaled with the size of a
single cell. The dimensionless form for the no void
condition (1) is
whereby using p = 1 -- 11 - h the equation for p can
be decoupled. The modelling is based on a continuum
of live cells which divide or die at rate depending on the concentrations of nutrient, cellular and
inhibitive material. We assume that the cell-cell contacts are sufficiently strong that the effects of random
motion and chemotaxis are negligible. The nutrient is
assumed to diffuse from the surface and to be consumed by the living cells, the timescales for these
processes being significantly shorter than the cell
birth rate. The inhibitive material is diffusive and
generated from the products of cell death; it can be
broken down by the living cells into cellular material.
+
D, a
7%
( r ) , (23)
Dl,
-
where the dimensionless diffusion coefficients of
cellular material Dl, and inhibitor Dh are assumed
constant. We note the Equation (23) can be derived
by summing Equations (20), (22) and the equivalent
equation for p and using (19). The dimensionless
functions a , h , k and 1 can be interpreted as
being the net rates of birth, volume change,
nutrient consumption and inhibitor production and are
given by
where c,, A, 6, p, t+b, 11, &, ,32 and rn, are all constant.
The mitotic rate k , and death rate kd functions
are described below. In brief, X is dimensionless
volume of cellular material consumed during mitosis,
6 is the ratio of dead cell and live cell volun~es,
AVASCULAR TUMOUR GROWTH
/ I is the proportion of inhibitive material in a dead
cell (1 5 S), %? is the rate of inhibitor breakdown
and v is the gain in volume of cellular material
during breakdown of the inhibitor (v :i1 ) . The
consumption rate expression k includes both nutrient
consumption by day-to-day cellular activity (3, > 0)
and an additional amount consumed during mitosis
(J2 > 0). The birth rate and death rate expressions
considered are
where BIA, c d , p ( , h,, 0 , P are all constants with
(T 5 1 and P
1. We note that B/A is the ratio of
the maximum death rate B to the maximum birth
rate A. The numerical solutions throughout this paper
assumed that all the exponents 171, = 1; however, the
general forms of these fi~nctionshave been included
here so the model proposed is a natural extension to
the model deccribed in Ward and King (1997).
The dimensionless initial and boundary conditions are
<
atr=O
i)c
dr
drz
dr
dlz
ar
- -- - - - - =
I' =
0,
where the mass transfer coefficients Qi, and Q,, and
the external concentrations of cellular material po and
inhibitor ho are all assumed constant.
173
We note that the model contains a large number
of dimensionless parameters. with little or no
experimental data being available for a number of
these; see Ward and King (1997, 1999a and 1999b)
for the determination of estimates of parameter
values.
Appendix B: Stability of the Minor branch
solutions for small spheroids
In this Appendix we seek solutions of the transient
model in the (non-physical) limit O + O in the
vicinity of steady-state solutions along the Minor
branch. The appropriate scaling for S is given
by (17) and we write S = O""* << 1. Following
an initial transient, on the timescale t = 0 ( 0 ~ " ~ ) ,
in which there is negligible growth, the transient
model equations give at leading order c
1. h ho,
12
1 - P O ho, 11 aor-13, where no = a ( l : p o . ho).
Noting that F ( S ) 4 / 3 s 2 , the leading order equation
for spheroid growth is then
-
- -
--
-
dS *
At
--
ry
Oos*
-
3
-
%I,"
-,
3 s 'l
-
where
k,,, = k,, (I ,yo, ho).
Defining
Sc
= (8kr1,0/u0)'13 O - " ' S ~ for uo > 0, so that S c
gives the leading order saturation size, it follows that
to leading order
where So* = S"(0). We note if we use F ( S ) = 1
instead, then the appropriate rescaling is S = OS*
and the appropriate form for S&, is S c = 6k,,,,,/ao,
resulting in S * = S; + (So* - ~ ; ) e " o ' i ~ ;the discussion below also applies to this case.
For a" > 0 (Regime I) the steady-state S, = S c
is unstable; if S; > SG, so the initial size is greater
than the steady-state, the spheroid will grow larger
at an exponential rate (until S becomes of O(1)). By
contrast, if So* < S& the spheroid becomes extinct at
,
by
time t -- t ~ given
for F ( S ) = 1 - 2/S - 4 / 3 s 2 . In the case of F ( S ) =
1 we instead have tc = 3In(S$/(Sc - S,"))/ao. For
174
a0
J. P. WARD AND J. R. KING
< 0 (Regime 11) the Minor branch solution tends
to steady-state of finite size as O -+ 0, and we
have S& < 0 in (27); the above analysis shows that
solutions with S," = O(1) die out in finite time, again
indicating the Minor branch to be unstable.
Acknowledgements
The first author gratefully acknowledges the support
of an EPSRC Earmarked Research Studentship and
the second a Nuffield Foundation Science Research
Fellowship.
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